Laws of friction

All the theories and studies on friction can be simplified into three main laws, which are valid in most cases:

First Law of Amontons

The frictional force is directly proportional to the normal load.

Second Law of Amontons

Friction is independent of the apparent area of contact.

Third Law of Coulomb

Dynamic friction is independent of the relative sliding speed.

Coulomb later found deviations from Amontons’ laws in some cases.^[6] In systems with significant nonuniform stress fields, Amontons’ laws are not satisfied macroscopically because local slip occurs before the entire system slides.^[19]

Static friction

Consider a block of a certain mass m, placed in a quiet position on a horizontal plane. If you want to move the block, an external force ${\displaystyle {\vec {F}}_{out}}$ must be applied, in this way we observe a certain resistance to the motion given by a force equal to and opposite to the applied force, which is precisely the static frictional force ${\displaystyle {\vec {F}}_{s.f.}}$.^[20]

By continuously increasing the applied force, we obtain a value such that the block starts instantly to move. At this point, also taking into account the first two friction laws stated above, it is possible to define the static friction force as a force equal in modulus to the minimum force required to cause the motion of the block, and the coefficient of static friction ${\displaystyle \mu }$ as the ratio of the static friction force ${\displaystyle {\vec {F}}_{s.f.}}$. and the normal force at block ${\displaystyle {\vec {N}}}$, obtaining

$${\displaystyle {\left|{\vec {F}}_{s.f.}\right|}\leq \mu {\left|{\vec {N}}\right|}}$$

Dynamic friction

Once the block has been put into motion, the block experiences a friction force with a lesser intensity than the static friction force ${\displaystyle {\vec {F}}_{s.f.}}$. The friction force during relative motion is known as the dynamic friction force ${\displaystyle {\vec {F}}_{d.f.}}$. In this case it is necessary to take into account not only the first two laws of Amontons, but also of the law of Coulomb, so as to be able to affirm that the relationship between dynamic friction force ${\displaystyle {\vec {F}}_{d.f.}}$, coefficient of dynamic friction k and normal force N is the following:

$${\displaystyle \left|{\vec {F}}_{d.f.}\right|=k{\left|{\vec {N}}\right|}}$$

Static and dynamic friction coefficient

Dynamic and static coefficient

At this point it is possible to summarize the main properties of the static friction coefficients ${\displaystyle \mu }$ and the dynamic one ${\displaystyle k}$.

These coefficients are dimensionless quantities, given by the ratio between the intensity of the friction force ${\displaystyle {\vec {F}}_{f}}$and the intensity of the applied load ${\displaystyle {\vec {W}}}$, depending on the type of surfaces that are involved in a mutual contact, and in any case, the condition is always valid such that: ${\displaystyle \mu >k}$.

Usually, the value of both coefficients does not exceed the unit and can be considered constant only within certain ranges of forces and velocities, outside of which there are extreme conditions that modify these coefficients and variables.

In systems with significant nonuniform stress fields, the macroscopic static friction coefficient depends on the external pressure, system size, or shape because local slip occurs before the system slides.^[19]

The following table shows the values of the static and dynamic friction coefficients for common materials:

Rolling friction

In the case of bodies capable of rolling, there is a particular type of friction, in which the sliding phenomenon, typical of dynamic friction, does not occur, but there is also a force that opposes the motion, which also excludes the case of static friction. This type of friction is called rolling friction. Now we want to observe in detail what happens to a wheel that rolls on a horizontal plane. Initially the wheel is immobile and the forces acting on it are the weight force ${\displaystyle m{\vec {g}}}$ and the normal force ${\displaystyle {\vec {N}}}$given by the response to the weight of the floor.

At this point the wheel is set in motion, causing a displacement at the point of application of the normal force which is now applied in front of the center of the wheel, at a distance b, which is equal to the value of the rolling friction coefficient. The opposition to the motion is caused by the separation of the normal force and the weight force at the exact moment in which the rolling starts, so the value of the torque given by the rolling friction force is

$${\displaystyle {{\vec {M}}_{r.f.}}={\vec {b}}\times m{\vec {g}}}$$

rolling friction

What happens in detail at the microscopic level between the wheel and the supporting surface is described in Figure, where it is possible to observe what is the behavior of the reaction forces of the deformed plane acting on an immobile wheel.

Rolling the wheel continuously causes imperceptible deformations of the plane and, once passed to a subsequent point, the plane returns to its initial state. In the compression phase the plane opposes the motion of the wheel, while in the decompression phase it provides a positive contribution to the motion.

The force of rolling friction depends, therefore, on the small deformations suffered by the supporting surface and by the wheel itself, and can be expressed as ${\displaystyle |{\vec {F}}_{r}|=b|{\vec {N}}|}$, where it is possible to express b in relation to the sliding friction coefficient ${\displaystyle \mu }$ as ${\textstyle b={\mu v \over r}}$, with r being the wheel radius.

The surfaces

Going even deeper, it is possible to study not only the most external surface of the metal, but also the immediately more internal states, linked to the history of the metal, its composition and the manufacturing processes undergone by the latter.

it is possible to divide the metal into four different layers:

1. Crystalline structure – basic structure of the metal, bulk interior form;
2. Machined layer – layer which may also have inclusions of foreign material and which derives from the processing processes to which the metal has been subjected;
3. Hardened layer – has a crystalline structure of greater hardness than the inner layers, thanks to the rapid cooling to which they are subjected in the working processes;
4. Outer layer or oxide layer – layer that is created due to chemical interaction with the metal's environment and from the deposition of impurities.

The layer of oxides and impurities (third body) has a fundamental tribological importance, in fact it usually contributes to reducing friction. Another fact of fundamental importance regarding oxides is that if you could clean and smooth the surface in order to obtain a pure "metal surface", what we would observe is the union of the two surfaces in contact. In fact, in the absence of thin layers of contaminants, the atoms of the metal in question, are not able to distinguish one body from another, thus going to form a single body if put in contact.

The origin of friction

Contact between surfaces is made up of a large number of microscopic regions, in the literature called asperities or junctions of contact, where atom-to-atom contact takes place. The phenomenon of friction, and therefore of the dissipation of energy, is due precisely to the deformations that such regions undergo due to the load and relative movement. Plastic, elastic, or rupture deformations can be observed:

• Plastic deformations – permanent deformations of the shape of the bumps;
• Elastic deformations – deformations in which the energy expended in the compression phase is almost entirely recovered in the decompression phase (elastic hysteresis);
• Break deformations – deformations that lead to the breaking of bumps and the creation of new contact areas.

The energy that is dissipated during the phenomenon is transformed into heat, thus increasing the temperature of the surfaces in contact. The increase in temperature also depends on the relative speed and the roughness of the material, it can be so high as to even lead to the fusion of the materials involved.

In friction phenomena, temperature is fundamental in many areas of application. For example, a rise in temperature may result in a sharp reduction of the friction coefficient, and consequently, the effectiveness of the brakes.

The cohesion theory

The adhesion theory states that in the case of spherical asperities in contact with each other, subjected to a ${\displaystyle {\vec {W}}}$ load, a deformation is observed, which, as the load increases, passes from an elastic to a plastic deformation. This phenomenon involves an enlargement of the real contact area ${\displaystyle A_{r}}$, which for this reason can be expressed as:

$${\displaystyle A_{r}={{\vec {W}} \over D}}$$

where D is the hardness of the material definable as the applied load divided by the area of the contact surface.

If at this point the two surfaces are sliding between them, a resistance to shear stress t is observed, given by the presence of adhesive bonds, which were created precisely because of the plastic deformations, and therefore the frictional force will be given by

$${\displaystyle {\vec {F}}_{a}=A_{r}{\vec {t}}}$$

At this point, since the coefficient of friction is the ratio between the intensity of the frictional force and that of the applied load, it is possible to state that

$${\displaystyle \mu ={t \over D}}$$

thus relating to the two material properties: shear strength t and hardness. To obtain low value friction coefficients ${\displaystyle \mu }$ it is possible to resort to materials which require less shear stress, but which are also very hard. In the case of lubricants, in fact, we use a substrate of material with low cutting stress t, placed on a very hard material.

The force acting between two solids in contact will not only have normal components, as implied so far, but will also have tangential components. This further complicates the description of the interactions between roughness, because due to this tangential component plastic deformation comes with a lower load than when ignoring this component. A more realistic description then of the area of each single junction that is created is given by

$${\displaystyle {A_{i}}^{2}=\left({\frac {{\vec {W}}_{i}}{D}}\right)^{2}+\alpha \left({\frac {{\vec {F}}_{i}}{D}}\right)^{2}}$$

with ${\displaystyle \alpha }$ constant and a "tangent" force ${\displaystyle {\vec {F}}_{i}}$ applied to the joint.

To obtain even more realistic considerations, the phenomenon of the third body should also be considered, i.e., the presence of foreign materials, such as moisture, oxides or lubricants, between the two solids in contact. A coefficient c is then introduced which is able to correlate the shear strength t of the pure "material" and that of the third body ${\displaystyle t_{t.b.}}$

$${\displaystyle t=c\cdot t_{t.b.}}$$

with 0 < c < 1.

By studying the behavior at the limits it will be that for c = 0, t = 0 and for c = 1 it returns to the condition in which the surfaces are directly in contact and there is no presence of a third body. Keeping in mind what has just been said, it is possible to correct the friction coefficient formula as follows:

$${\displaystyle \mu ={\frac {c}{[\alpha (1-c^{2})]^{1/2}}}}$$

In conclusion, the case of elastic bodies in interaction with each other is considered.

Similarly to what we have just seen, it is possible to define an equation of the type

$${\displaystyle A=K{\vec {W}}}$$

where, in this case, K depends on the elastic properties of the materials. Also for the elastic bodies the tangential force depends on the coefficient c seen above, and it will be

$${\displaystyle {\vec {F}}_{T}=cAs}$$

and therefore a fairly exhaustive description of the friction coefficient can be obtained

$${\displaystyle \mu =cKs}$$

Friction measurements

The simplest and most immediate method for evaluating the friction coefficient of two surfaces is the use of an inclined plane on which a block of material is made to slide. As can be seen in the figure, the normal force of the plane is given by ${\displaystyle mg\cos \theta }$, while the frictional force is equal to ${\displaystyle mg\sin \theta }$. This allows us to state that the coefficient of friction can be calculated very easily, by means of the tangent of the angle in which the block begins to slip. In fact we have

$${\displaystyle \mu ={F_{a} \over N}={mg\sin \theta \over mg\cos \theta }={\sin \theta \over \cos \theta }=\tan \theta }$$

Then from the inclined plane we moved on to more sophisticated systems, which allow us to consider all the possible environmental conditions in which the measurement is made, such as the cross-roller machine or the pin and disk machine. Today there are digital machines such as the "Friction Tester" which allows, by means of a software support, to insert all the desired variables. Another widely used process is the ring compression test. A flat ring of the material to be studied is plastically deformed by means of a press, if the deformation is an expansion in both the inner and the outer circle, then there will be low or zero friction coefficients. Otherwise for a deformation that expands only in the inner circle there will be increasing friction coefficients.

Viscosity

The viscosity is the equivalent of friction in fluids, it describes, in fact, the ability of fluids to resist the forces that cause a change in shape.

Thanks to Newton's studies, a deeper understanding of the phenomenon has been achieved. He, in fact, introduced the concept of laminar flow: "a flow in which the velocity changes from layer to layer". It is possible to ideally divide a fluid between two surfaces (${\displaystyle S_{1}}$, ${\displaystyle S_{2}}$) of area A, in various layers.

The layer in contact with the surface ${\displaystyle S_{2}}$, which moves with a velocity v due to an applied force F, will have the same velocity as v of the slab, while each immediately following layer will vary this velocity of a quantity dv, up to the layer in contact with the immobile surface ${\displaystyle S_{1}}$, which will have zero speed.

From what has been said, it is possible to state that the force F, necessary to cause a rolling motion in a fluid contained between two plates, is proportional to the area of the two surfaces and to the speed gradient:

$${\displaystyle F\propto A{dv \over dy}}$$

At this point we can introduce a proportional constant ${\displaystyle \mu }$, which corresponds to the dynamic viscosity coefficient of the fluid, to obtain the following equation, known as Newton's law

$${\displaystyle F=\mu A{dv \over dy}}$$

The speed varies by the same amount dv of layer in layer and then the condition occurs so that dv / dy = v / L, where L is the distance between the surfaces ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$, and then we can simplify the equation by writing

$${\displaystyle F=\mu A{v \over L}}$$

The viscosity ${\displaystyle \mu }$ is high in fluids that strongly oppose the motion, while it is contained for fluids that flow easily.

To determine what kind of flow is in the study, we observe its Reynolds number

$${\displaystyle Re={{\rho Lv} \over \mu }}$$

This is a constant that depends on the fluid mass ${\displaystyle \rho }$ of the fluid, on its viscosity ${\displaystyle \mu }$ and on the diameter L of the tube in which the fluid flows. If the Reynolds number is relatively low then there is a laminar flow, whereas for ${\displaystyle Re\simeq 2000}$ the flow becomes turbulent.

To conclude we want to underline that it is possible to divide the fluids into two types according to their viscosity:

1. Newtonian fluids, or fluids in which viscosity is a function of temperature and fluid pressure only and not of velocity gradient;
2. Non-Newtonian fluids, or fluids in which viscosity also depends on the velocity gradient.

Viscosity as a function of temperature and pressure

Temperature and pressure are two fundamental factors to evaluate when choosing a lubricant instead of another. Consider the effects of temperature initially.

There are three main causes of temperature variation that can affect the behavior of the lubricant:

• Weather conditions;
• Local thermal factors (like for car engines or refrigeration pumps);
• Energy dissipation due to rubbing between surfaces.

In order to classify the various lubricants according to their viscosity behavior as a function of temperature, in 1929 the viscosity index (V.I.) was introduced by Dean and Davis. These assigned the best lubricant then available, namely the oil of Pennsylvania, the viscosity index 100, and at the worst, the American oil of the Gulf Coast, the value 0. To determine the value of the intermediate oil index, the following procedure is used: two reference oils are chosen so that the oil in question has the same viscosity at 100 °C, and the following equation is used to determine the viscosity index

$${\displaystyle V.I.={{L-O_{\text{Test}}} \over {L-H}}\times 100}$$

This process has some disadvantages:

• For mixtures of oils the results are not exact;
• There is no information if you are outside the fixed temperature range;
• With the advancement of the technologies, oils with V.I. more than 100, which can not be described by the method above.

In the case of oils with V.I. above 100 you can use a different relationship that allows you to get exact results

$${\displaystyle V.I.={{10^{N-1}} \over {0.00715}}+100}$$

$${\displaystyle N={{\operatorname {Log} (H)-\operatorname {Log} (O_{\text{Test}})} \over {\operatorname {Log} (v)}}}$$

where, in this case, H is the viscosity at 100 °F (38 °C) of the oil with V.I. = 100 and v is the kinematic viscosity of the study oil at 210 °F (99 °C).

We can therefore say, in conclusion, that an increase in temperature leads to a decrease in the viscosity of the oil. It is also useful to keep in mind that, in the same way, an increase in pressure implies an increase in viscosity. To evaluate the effects of pressure on viscosity, the following equation is used

$${\displaystyle \mu =\mu _{0}\exp {(\alpha p)}}$$

where ${\displaystyle \mu }$ is the pressure viscosity coefficient p, ${\displaystyle \mu _{0}}$ is the viscosity coefficient at atmospheric pressure and ${\displaystyle \alpha }$ is a constant that describes the relationship between viscosity and pressure.

Viscosity measures

To determine the viscosity of a fluid, viscosimeters are used which can be divided into 3 main categories:

• Capillary viscometers, in which the viscosity of the fluid is measured by sliding it into a capillary tube;
• Solid drop viscometers, in which viscosity is measured by calculating the velocity of a solid that moves in the fluid;
• Rotational viscometers, in which viscosity is obtained by evaluating the flow of fluid placed between two surfaces in relative motion.

The first two types of viscometers are mainly used for Newtonian fluids, while the third is very versatile.

Adhesive wear

As known, the contact between two surfaces occurs through the interaction between asperities.^[22] If a shearing force is applied in the contact area, it may be possible to detach a small part of the weaker material, due to its adhesion to the harder surface. What is described is precisely the mechanism of the adhesive wear represented in the figure. This type of wear is very problematic, since it involves high wear speeds, but at the same time it is possible to reduce adhesion by increasing surface roughness and hardness of the surfaces involved, or by inserting layers of contaminants such as oxygen, oxides, water, or oils. In conclusion, the behavior of the adhesive wear volume can be described by means of three main laws

Law 1 – Distance

The mass involved in wear is proportional to the distance traveled in the rubbing between the surfaces.

Law 2 – Load

The mass involved in wear is proportional to the applied load.

Law 3 – Hardness

The mass involved in wear is inversely proportional to the hardness of the less hard material.

An important aspect of wear is emission of wear particles into the environment which increasingly threatens human health and ecology. The first researcher who investigated this topic was Ernest Rabinowicz.^[23]

Abrasive wear

The abrasive wear consists of the cutting effort of hard surfaces that act on softer surfaces and can be caused either by the roughness that as tips cut off the material against which they rub (two-body abrasive wear), or from particles of hard material that interpose between two surfaces in relative motion (three-body abrasive wear). At application levels, the two-body wear is easily eliminated by means of an adequate surface finish, while the three-body wear can bring serious problems and must therefore be removed as much as possible by means of suitable filters, even before of a weighted machine design.

Fatigue wear

The fatigue wear is a type of wear that is caused by alternative loads, which cause local contact forces repeated over time, which in turn lead to deterioration of the materials involved. The most immediate example of this type of wear is that of a comb. If you slide a finger over the teeth of the comb over and over again, it is observed that at some point one or more teeth of the comb come off. This phenomenon can lead to the breaking of the surfaces due to mechanical or thermal causes. The first case is that described above in which a repeated load causes high contact stresses. The second case, however, is caused by the thermal expansion of the materials involved in the process. To reduce this type of wear, therefore, it is good to try to decrease both the contact forces and the thermal cycling, that is the frequency with which different temperatures intervene. For optimal results it is also good to eliminate, as much as possible, impurities between surfaces, local defects and inclusions of foreign materials in the bodies involved.

Corrosive wear

The corrosive wear occurs in the presence of metals that oxidize or corrode. When the pure metal surfaces come into contact with the surrounding environment, oxide films are created on their surfaces because of the contaminants present in the environment itself, such as water, oxygen or acids. These films are continually removed from the abrasive and adhesive wear mechanisms, continually recreated by pure-contaminating metal interactions. Clearly this type of wear can be reduced by trying to create an 'ad hoc' environment, free of pollutants and sensible to minimal thermal changes. Corrosive wear can also be positive in some applications. In fact, the oxides that are created, contribute to decrease the coefficient of friction between the surfaces, or, being in many cases harder than the metal to which they belong, can be used as excellent abrasives.

Rubbing wear or fretting

The rubbing wear occurs in systems subject to more or less intense vibrations, which cause relative movements between the surfaces in contact within the order of nanometers. These microscopic relative movements cause both adhesive wear, caused by the displacement itself, and abrasive wear, caused by the particles produced in the adhesive phase, which remain trapped between the surfaces. This type of wear can be accelerated by the presence of corrosive substances and the increase in temperature.^[24]

Erosion wear

The erosion wear occurs when free particles, which can be either solid or liquid, hit a surface, causing abrasion. The mechanisms involved are of various kinds and depend on certain parameters, such as the impact angle, the particle size, the impact velocity and the material of which the particles are made up.

Factors affecting wear

Among the main factors influencing wear we find

• Hardness
• Mutual Solubility
• Crystalline structure

It has been verified that the harder a material is, the more it decreases. In the same way, the less two materials are mutually soluble, the more the wear tends to decrease. Finally, as regards the crystalline structure, it is possible to state that some structures are more suitable to resist the wear of others, such as a hexagonal structure with a compact distribution, which can only deform by slipping along the base planes.

Wear rate

To provide an assessment of the damage caused by wear, we use a dimensionless coefficient called wear rate, given by the ratio between the height change of the body ${\displaystyle \Delta h}$ and the length of the relative sliding ${\displaystyle \Delta l}$.

$${\displaystyle T_{\text{wear}}={\delta h \over \Delta l}}$$

This coefficient makes it possible to subdivide, depending on its size, the damage suffered by various materials in different situations, passing from a modest degree of wear, through a medium, to a degree of severe wear.

Instead, to express the volume of wear V it is possible to use the Holm equation

• ${\displaystyle V=k{W \over H}l}$ (for adhesive wear)
• ${\displaystyle V=k_{a}{W \over H}l}$ (for abrasive wear)

where W / H represents the real contact area, l the length of the distance traveled and k and ${\displaystyle k_{a}}$ are experimental dimensional factors.

Wear measurement

In experimental measurements of material wear, it is often necessary to recreate fairly small wear rates and to accelerate times. The phenomena, which in reality develop after years, in the laboratory must occur after a few days. A first evaluation of the wear processes is a visual inspection of the superficial profile of the body in the study, including a comparison before and after the occurrence of the wear phenomenon. In this first analysis the possible variations of the hardness and of the superficial geometry of the material are observed. Another method of investigation is that of the radioactive tracer, used to evaluate wear at macroscopic levels. One of the two materials in contact, involved in a wear process, is marked with a radioactive tracer. In this way, the particles of this material, which will be removed, will be easily visible and accessible. Finally, to accelerate wear times, one of the best-known techniques used is that of the high pressure contact tests. In this case, to obtain the desired results it is sufficient to apply the load on a very reduced contact area.